American Institute of Mathematical Sciences

Advanced
February  2010, 27(1): 205-216. doi: 10.3934/dcds.2010.27.205

$C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings

Shaobo Gan 1, , Kazuhiro Sakai 2, and  Lan Wen 3,

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871
2. 

Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505
3. 

School of Mathematic Sciences, Peking University, Beijing, 100871

Received  June 2009 Revised  November 2009 Published  February 2010

Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$. In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting. Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.
Keywords: dominated splitting, partially hyperbolic., shadowing, Weak shadowing, pseudo-orbit, homoclinic class, chain component, chain recurrent set.
Mathematics Subject Classification: Primary: 37B20, 37C29, 37C50, 37D20, 37D3.
Citation: Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205
[1]

Flavio Abdenur, Lorenzo J. Díaz. Pseudo-orbit shadowing in the $C^1$ topology. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 223-245. doi: 10.3934/dcds.2007.17.223
[2]

Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107
[3]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[4]

S. Yu. Pilyugin, A. A. Rodionova, Kazuhiro Sakai. Orbital and weak shadowing properties. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 287-308. doi: 10.3934/dcds.2003.9.287
[5]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[6]

Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019
[7]

Sergei Yu. Pilyugin. Variational shadowing. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733
[8]

S. Yu. Pilyugin, Kazuhiro Sakai, O. A. Tarakanov. Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 871-882. doi: 10.3934/dcds.2006.16.871
[9]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[10]

Keonhee Lee, Kazuhiro Sakai. Various shadowing properties and their equivalence. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 533-540. doi: 10.3934/dcds.2005.13.533
[11]

Shaobo Gan. A generalized shadowing lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 627-632. doi: 10.3934/dcds.2002.8.627
[12]

Sergey V. Bolotin. Shadowing chains of collision orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 235-260. doi: 10.3934/dcds.2006.14.235
[13]

S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29
[14]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[15]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[16]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[17]

Will Brian, Jonathan Meddaugh, Brian Raines. Shadowing is generic on dendrites. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2211-2220. doi: 10.3934/dcdss.2019142
[18]

Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629
[19]

Suzanne Lynch Hruska. Rigorous numerical models for the dynamics of complex Hénon mappings on their chain recurrent sets. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 529-558. doi: 10.3934/dcds.2006.15.529
[20]

Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences